Why the name linear fractional map?

Question

Fractional linear transformation is a map from extended complex plane to itself, defined by: \begin{equation} z\to \frac{az+b}{cz+d} \end{equation} with $ad-bc\ne0$.

Wikipedia says that "a linear fractional transformation is a transformation that is represented by a fraction whose numerator and denominator are linear". Either in the numerator or denominator there are translations and translations aren't linear transformation (they can't map origin in the origin). So, why "linear" in the name?

I know that for an inversion:

Is for this reason the name?

Answer 1

Functions of the form $f(x) = mx+b$ are often called "linear" because their graphs are lines. This usage differs from the more restrictive notion of a linear function $F$ being one satisfying $F(\alpha x+\beta y)=\alpha F(x)+\beta F(y)$.

Answer 2

I am not a historian, but: The Riemann sphere may be identified with the complex projective line, the set of complex lines through the origin in the "complex plane" $\mathbf{C}^{2}$. Fractional linear transformations on the sphere are precisely those induced by invertible complex-linear transformations.

Geometrically, the invertible linear transformation with standard matrix $\left[\begin{array}{@{}cc@{}}a & b \\ c & d \\ \end{array}\right]$ on the (complex) Cartesian plane acts on the set of lines through the origin. Viewed as a mapping on slopes, the line of slope $z$ maps to the line of slope $\dfrac{az + b}{cz + d}$. Algebraically, let $(Z, W)$ denote Cartesian coordinates, and write $z = Z/W$. Up to scaling, i.e., in projective coordinates, the point $$ \left[\begin{array}{@{}c@{}} z \\ 1 \\ \end{array}\right] = \left[\begin{array}{@{}c@{}} Z/W \\ 1 \\ \end{array}\right] \simeq \left[\begin{array}{@{}c@{}} Z \\ W \\ \end{array}\right] $$ maps to $$ \left[\begin{array}{@{}cc@{}}a & b \\ c & d \\ \end{array}\right] \left[\begin{array}{@{}c@{}} Z \\ W \\ \end{array}\right] = \left[\begin{array}{@{}c@{}} aZ + bW \\ cZ + dW \\ \end{array}\right] \simeq \left[\begin{array}{@{}c@{}} \dfrac{aZ + bW}{cZ + dW} \\ 1 \\ \end{array}\right] = \left[\begin{array}{@{}c@{}} \dfrac{az + b}{cz + d} \\ 1 \\ \end{array}\right]. $$

As for where the name fractional linear transformation originated, it seems in line with 19th century geometry that "linear" connotes action on the Cartesian plane and "fractional" refers to extracting a slope from points on a line through the origin. But to reiterate, I am not a historian.